Wikipedia / Chebotarev density theorem
Wikipedia / Chebotarev density theorem
In mathematics, specifically in algebraic number theory, the Chebotarev density theorem, named after Nikolai Chebotarev, statistically describes the splitting of primes in a given Galois extension of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers goes to infinity. It was proved by Chebotarev in his thesis in 1922.
is an algebraic number field among all primes. More generally, splitting behavior can be specified by assigning to (almost) every prime number an invariant, its Frobenius element, which is a representative of a well-defined conjugacy class in the Galois group.
History and motivation
When Carl Friedrich Gauss first introduced the notion of complex integers is 2 then it becomes a product of the square of the prime and the invertible Gaussian integer ; we say that 2 "ramifies". For instance,
splits completely;
is inert;
ramifies.
. Dirichlet's theorem on arithmetic progressions follows a simple statistical law.
Similar statistical laws also hold for splitting of primes in the cyclotomic extensions, obtained from the field of rational numbers by adjoining a primitive root of unity, according to their pattern of splitting in the ring of integers corresponding to the 8th roots of unity. In this case, the field extension has degree 4 and is abelian, with the Galois group isomorphic to the Klein four-group. It turned out that the Galois group of the extension plays a key role in the pattern of splitting of primes. Georg Frobenius established the framework for investigating this pattern and proved a special case of the theorem. The general statement was proved by Nikolai Grigoryevich Chebotaryov in 1922.
Relation with Dirichlet's theorem
The Chebotarev density theorem may be viewed as a generalisation of Dirichlet's theorem on arithmetic progressions. A quantitative form of Dirichlet's theorem states that if is coprime to mod is asymptotic to , where is the Euler totient function.
This is a special case of the Chebotarev density theorem for the -th cyclotomic field . Indeed, the Galois group of is abelian and can be canonically identified with the group of invertible residue classes mod . The splitting invariant of a prime not dividing splits is , where is multiplicative order modulo ; hence, by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to .
Formulation
In their survey article, is a splitting field is a Galois extension factorizes in some fashion over the prime field. If , then the splitting type is a partition of ; in other words, by choosing an ordering of a root and its algebraic conjugates is faithfully represented as a subgroup of the symmetric group by means of its cycle representation, which gives a "cycle type" , again a partition of .
The theorem of Frobenius states that for any given choice of is has a natural density that have cycle type .
The statement of the more general Chebotarev density theorem is in terms of the Frobenius element of a prime (ideal), which is in fact an associated conjugacy class of elements of the Galois group is abelian, the classes each have size 1. For the case of a non-abelian group with it as Galois group.
Statement
. The statement is valid when the density refers to either the natural density or the analytic density of the set of primes.
Effective version
The generalized Riemann hypothesis implies an effective version is
where the constant implied in the big-O notation its discriminant.
to be the Artin conductor,
is an exceptional real zero term can be ignored. The implicit constant of this expression is absolute.
Infinite extensions
that is unramified outside a finite set is a profinite group is compact in this topology, there is a unique Haar measure. The Chebotarev density theorem in this situation can be stated as follows:
is finite (the Haar measure is then just the counting measure.
Important consequences
that split completely in it..
See also
Notes
References